k  এর মানের জন্য ⁿp_r=k(ⁿ⁺¹C_r -ⁿC_r-1)  হবে?

Explanation:

Another Explanation (5): প্রশ্ন: k এর মানের জন্য ⁿP_r=k(ⁿ⁺¹C_r -ⁿC_r-1) হবে? সমাধান: আমরা জানি, ⁿP_r = \(\frac{n!}{(n-r)!}\) এবং ⁿC_r = \(\frac{n!}{r!(n-r)!}\) এখন, ⁿ⁺¹C_r - ⁿC_r-1 = \(\frac{(n+1)!}{r!(n+1-r)!} - \frac{n!}{(r-1)!(n-r+1)!}\) = \(\frac{(n+1)!}{r!(n+1-r)!} - \frac{n!}{(r-1)!(n-r+1)!}\) = \(\frac{(n+1)n!}{r(r-1)!(n+1-r)(n-r)!} - \frac{n!}{(r-1)!(n-r+1)(n-r)!}\) = \(\frac{n!}{(r-1)!(n-r)!} [\frac{n+1}{r(n+1-r)} - \frac{1}{n-r+1}]\) = \(\frac{n!}{(r-1)!(n-r)!} [\frac{n+1 -r}{r(n+1-r)}] \) = \(\frac{n!}{(r-1)!(n-r)!} [\frac{1}{r}] \) = \(\frac{n!}{r(r-1)!(n-r)!}\) = \(\frac{n!}{r!(n-r)!}\) = \(\frac{1}{r!} \cdot \frac{n!}{(n-r)!}\) = \(\frac{1}{r!} \cdot \) ⁿP_r সুতরাং, ⁿP_r=k(ⁿ⁺¹C_r -ⁿC_r-1) implies, ⁿP_r = k \(\cdot \frac{1}{r!}\) ⁿP_r অতএব, k = r! উত্তর: r! 🎉